# A numerical scheme for stochastic differential equations with distributional drift

## TL;DR

This paper introduces a novel numerical scheme for solving one-dimensional stochastic differential equations with distributional drifts in fractional Sobolev spaces, providing convergence rates and practical implementation.

## Contribution

It is the first to develop and implement a numerical method for SDEs with distributional drifts, extending the scope of stochastic numerical analysis.

## Key findings

- Established convergence rate in $L^1$-norm for the scheme.
- Successfully implemented the numerical scheme.
- Provided convergence estimates for drifts in $L^p$-spaces.

## Abstract

In this paper we present a scheme for the numerical solution of one-dimensional stochastic differential equations (SDEs) whose drift belongs to a fractional Sobolev space of negative regularity (a subspace of Schwartz distributions). We obtain a rate of convergence in a suitable $L^1$-norm and we implement the scheme numerically. To the best of our knowledge this is the first paper to study (and implement) numerical solutions of SDEs whose drift lives in a space of distributions. As a byproduct we also obtain an estimate of the convergence rate for a numerical scheme applied to SDEs with drift in $L^p$-spaces with $p\in(1,\infty)$.

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Source: https://tomesphere.com/paper/1906.11026